Determine the locus of the point of intersection of the altitudes (orthocenter) of a triangle , if the locus of the vertex is a line parallel to .
Let denote the interior angle at A, then
1. the parabola cuts the parallel to BC through A in 2 points if ;
2. the parallel to BC through A is tangent to the parabola if ;
3. the vertex of the parabola is located between the parallel to Bc and the line BC if ;
Place the base BC of the triangle on the x-axis. Let d denote the distance between the 2 parallels. B(0,0) and C(c,0) and A(x,d).
The slope of AC is . Then the height through B (perpendicular to AC) has the slope:
Then the orthocenter is
That means all orthocenters lie on the curve of
I've choosen d = 5, C(8,0)